Download Lives of the Stars Lecture 3: What makes a star?

Lives of the Stars Lecture 3:
What makes a star?
Presented by
Dr Helen Johnston
School of Physics
Spring 2016
The University of Sydney
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Night-viewing evening
The night-viewing evening, scheduled for Saturday 29 October, has
been
CANCELLED
John will try for a future date. If you would like to be notified of this,
please leave your name and email address on the sheet at the front.
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Prologue
What is a star?
“... a gravitationally confined thermonuclear reactor whose
composition evolves as energy is lost to radiation and neutrinos”
– Woosley, Heger and Weaver 2002
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Prologue
What makes a star?
In fact, there is a simple answer to this question:
Gravity
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To understand why, we need to look at the four fundamental forces
which govern the universe.
Forces are the means by which particles in the universe interact with
each other: no evidence for a fifth force has ever been found.
Every particle in the Universe reacts to at least one of these forces.
Forces can be either attractive or repulsive.
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Force
acts on
strong nuclear
protons
neutrons
electromagnetic
Strength Range
1
10–15 m
charged particles:
protons, electrons
0.007
infinite
weak nuclear
protons, neutrons,
electrons, neutrinos
10–5
10–17 m
gravitational
particles with mass:
protons, neutrons,
electrons
10–39
infinite
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• The strong nuclear force is the strongest, but falls off very rapidly
with distance: it only acts over the diameter of the nucleus
• The weak nuclear force has an even shorter range than the strong
force, and governs processes like beta decay and radioactivity.
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• The electromagnetic force governs all interactions between protons
and electrons. It falls off as the square of the distance, so its range is
infinite.
The direction of the force depends on the sign of the charges.
Opposite charges attract
–
+
Like charges repel
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+
+
–
–
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• Gravity acts on all particles with mass. Like the electromagnetic force,
it falls off as the square of the distance, so its range is infinite.
However, unlike the electromagnetic force, there is no repelling
component: there is no “negative mass”. Gravity always attracts.
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The EM force is incomparably stronger than the gravitational force. If
you put a person in orbit and left 1% of their electrons behind, the EM
force would be strong enough to affect the orbit of the Earth!
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So why do we rarely have to worry about the EM force in astronomy?
The Earth contains about 1051 protons, but it also contains about 1051
electrons. This means that the overall electric charge on the Earth is
quite close to zero.
The gravitational field of the Earth, however, is 1051 times that of a
single proton: the fields of each and every particle that makes up the
Earth all add up to make the overall gravitational field of the Earth.
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That is why the gravitational force is so important in the universe. All the
other forces are essentially only local, whereas the gravitational force is
both long-range and only attractive.
This is why gravity eventually wins.
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Stars
Let’s see how these facts about gravity affect the behaviour of stars.
A star consists of about 2x1030 kg of gas, mostly hydrogen and
helium.
If each of those atoms is attracting each of the others, what is
stopping the gas collapsing in on itself?
Something must be resisting the collapse.
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The gas itself is what resists the collapse. As the outer layers press
down on the gas in the interior, the pressure increases: the individual
atoms are squeezed together.
The gas is made of individual particles, and the behaviour we observe
of gases at large scale (macroscopic quantities) relate to the small-scale
behaviour of these particles (microscopic quantities).
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Gases are made up of tiny particles moving in straight lines and
bouncing off each other – the kinetic theory of gases.
High temperature means particles are moving faster (more kinetic
energy); high pressure is the transfer of kinetic energy from particles
hitting the sides of the container
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In air at 0o C, the molecules are
typically travelling at 400 m/s, though
they have a spread of speeds from
almost zero up to 1200 m/s. However,
they never get very far: on average, a
particle only travels 2x10–5 cm before
colliding with another molecule, so on
average each molecule makes five
billion collisions per second!
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Everything we know about the behaviour of gases follows from this
simple model.
This behaviour summarised in the ideal gas law:
PV = nRT
as P goes up:
V goes down, T up, density up
as V goes down: P goes up, T up, density up
as T goes up:
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P goes up, V up, density down
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So: gravity is squeezing the gas in the star. This increases the pressure,
and also increases the temperature of the gas at the centre.
In fact, we can work out very roughly what the temperature in the
middle of the star is, by working out what pressure is required to
counterbalance gravity. A very rough estimate
shows that the temperature must be about
weight
20,000,000 K.
pushing
down
pressure
pushing
up T
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Let’s leave this for a moment, and consider another interesting fact
about stars: they are shining, i.e. losing energy.
Stars must be producing energy, since they can keep this up for billions
of years.
Where do they get this energy from?
Chemical energy?
only enough for a million years
Gravitational energy?
only enough for 100 million years
Since we have good evidence that the Solar System is at least 4 billion
years old, we need another source.
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The only possible source of energy which could keep the Sun burning
for billions of years is nuclear energy: somehow, the Sun is converting a
tiny fraction of its mass into energy according to Einstein’s relationship
E = mc2
Since stars are largely hydrogen, they provide this energy by the
simplest method: fusing four hydrogen atoms together to form helium.
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At the enormous temperatures the star’s core, ordinary materials cannot
exist as solids or liquids. Instead, the atoms are all completely ionised,
so the bare nuclei of the atoms are moving freely through a sea of
electrons: the star’s interior is a plasma, the fourth state of matter.
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Freed from their repelling electrons, the nuclei can now approach much
closer to each other until, at high enough temperatures, they can
overcome the electromagnetic repulsion and get close enough for the
strong nuclear force to fuse them together.
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The nuclei get their speed from the random motion of the gas, so fusion
only occurs when the temperature is high enough.
The temperature threshold for fusion to occur is about ten million
degrees.
One consequence is that fusion only takes place in the cores of stars,
where the temperature is highest. Most of the star is not in fact
generating energy. This is also, of course, why fusion
is not taking place in the core of the Earth
(temperature 6000K) or of Jupiter
(temperature 20,000 K).
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Aside:
In fact, the required temperature should be even higher than ten
million degrees, except for the quantum mechanical effect called
quantum tunnelling. Because each particle also behaves as a wave,
there is a small probability that it can tunnel through the barrier even
when it doesn’t have enough energy to go over.
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Classically, if the proton does not have enough energy to overcome the
barrier of the electromagnetic repulsion, it will be reflected.
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A quantum particle, on the other hand, has a finite probability of
tunnelling through the barrier.
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The reaction forming helium from hydrogen actually takes four hydrogen
atoms, and takes place in several steps.
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One helium atom weighs slightly less than four hydrogen atoms: 0.7%
less. This missing mass is converted into energy, and is what powers the
star. (This is the binding energy we discussed last week).
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We can work out how much mass is being converted to energy each
second: knowing the luminosity of the Sun, for instance (3.8 x 1026 W),
then the amount of mass lost each second is L/c2, which comes to
4,200,000,000 kg every second.
Which sounds like a lot, except that the mass of the Sun is 2 x 1030 kg,
so even after a billion years, the Sun has only lost 0.006% of its mass.
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Hydrogen can also be fused to helium in a different sequence of steps:
the CNO cycle.
The carbon atom is used as
a catalyst, but is not
consumed during the
reaction, so reappears at
the end ready to start the
cycle over again.
The CNO cycle requires
much higher temperatures
than the p–p chain.
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Hydrogen fusion will only take place above temperatures of about ten
million degrees, and the rate of fusion reactions increases dramatically
as the temperature increases: the rate is ∝ T4 (or T20 for the CNO cycle
in hot stars).
One consequence of this is that the central temperature of stars doesn’t
vary very much: a 100 solar mass star has a central temperature only a
factor of 4 higher than a 0.1 solar mass red dwarf.
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This provides the thermostat which maintains the star in pressure
equilibrium. If the pressure in the core increases, the temperature
increases, which increases the fusion reaction rate, which would produce
more energy and increase the temperature, which would expand the
star, until the pressure drops again.
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This also explains the lower limit to a star’s mass. If the collapsing star is
too small, the central temperature and density never get high enough
for fusion to take place, so the almost-star never ignites.
The lowest mass require to produce fusion is about 0.08 solar masses, or
about 80 Jupiter masses. Objects smaller than this are called brown
dwarfs.
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Discovery image of the first confirmed brown dwarf,
Gliese 229B; artist’s impression of the system.
There is also an upper limit to the mass a star can have, but this is
because of radiation pressure.
Because a photon is also a particle, the photons can also transfer
momentum to particles: they add to the pressure.
For the centre of the Sun,
Prad = 0.06% of Pgas
But for a star of mass 60 times solar,
Prad = Pgas
When the star’s mass reaches about 100 solar masses, the radiation
pressure becomes high enough to blow off the outer layers of the star.
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Hubble picture of the star Eta Carinae,
one of the most massive stars in the
Galaxy. The lobes are remnants from
The
University
Sydney
the
starof ejecting
its outer layers.
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So a star is a finally balanced system:
gravity
produces
which balances
high pressure
high temperature
which produces
which produces
higher temperature
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nuclear fusion
which produces
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That’s the general outline. We can actually find out the exact details of
a star’s interior by mathematical modelling: Divide the star into very
narrow shells, like an onion. Then we can write down equations showing
the density, temperature and pressure in each one of these shells, so
when we put them all together we have a star.
gravity pushing
inwards
pressure pushing
outwards
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Now, there are four simple laws which must be satisfied:
• Conservation of mass: total mass equals the sum of shell masses
• Conservation of energy: total luminosity equals the sum of energy
generated in each shell
• Hydrostatic equilibrium: the outward pressure in each shell balances
the inward gravitational pull on that shell
• Energy transport: energy moves from hot to cool by conduction,
radiation or convection
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We solve these equations simultaneously for each shell in the star, to
give us a stellar model: a model for how the density, mass and
temperature change as we move from the inside to the outside of the
star.
It takes a computer to solve the hundreds of simultaneous equations
needed for a typical model.
An example on the web:
http://www.astro.umass.edu/~weinberg/a451/msapplet.html
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Results of stellar modelling
What we get out of the model is a description of the conditions inside
the star: the temperature, density, mass and luminosity as a function of
radius.
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What can we learn from stellar models?
1. Mass-luminosity relation
As we change the mass of the star, the luminosity of the star (= the
energy that reaches the surface) changes in a systematic way: the
more massive a star is, the more luminous it is.
More mass → higher pressure in centre
→ higher temperature in centre
→ more energy produced
→ higher luminosity
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As we increase the mass of the star, the luminosity increases enormously:
a factor of 10 increase in mass corresponds to a factor of 3000
increase in luminosity.
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2. Fusion occurs in the central regions
We saw how we need temperatures of at
least 10 million degrees for fusion to
occur.
In the Sun, these temperatures are
reached in the inner 25% of the radius, in
the zone known as the core.
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Complications: Convection
There are complications to the
structure of stars that we haven’t
touched on. The interior of a star is
divided into different zones,
depending on whether energy is
being transported by radiation or
convection.
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Convection cells visible on the Sun’s surface as granularity
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Whether the energy is transferred by radiation or convection depends
on the opacity of the envelope: how well the material lets radiation
through.
If the temperature is high and the atoms are all ionised, the opacity is
low and the energy streams out by radiation. As the temperature drops,
the protons and electrons recombine, and the electrons start absorbing
photons, which slows down the transport. This traps the heat, leading to
bubbles of hot material, which begin to rise and start convecting.
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In high-mass stars, the hydrogen stays ionised even in the outer regions,
so radiation is not easily absorbed: high mass stars have radiative
envelopes.
In their cores, however, so much energy is being produced that radiative
transport doesn’t work, so high mass stars have convective cores.
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Photons don’t reach the surface of the sun unimpeded. Instead, they
undergo many scatterings on the way and only slowly diffuse outwards,
taking several thousand years to reach the surface: the photosphere.
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Testing the theory of stellar fusion: The solar neutrino
problem
We can’t see fusion taking place in the cores of stars, but we can
detect it using neutrinos: sub-atomic particles which have no mass or
charge, and which are produced as by-products of H → He fusion.
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Neutrinos have a very low probability of interacting with normal matter,
so they escape from the core of the Sun almost directly.
Photons take a
tortuous path out of
the Sun, taking
thousands of years to
get out
Neutrinos zip out in
about 2 seconds
However, it is possible to detect them at the Earth. The number detected
should be proportional to the number of fusions taking place.
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Raymond Davis built the first neutrino
detector in the 1960s. The detector
consisted of 400,000 litres of
cleaning fluid (C2Cl4) one mile
underground in an old mine.
When a chlorine atom captures a
neutrino, it is turned into a
radioactive isotope of argon. A few
dozen events are expected every
month.
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The results of the experiment were astonishing. Davis detected only
about a third of the expected number of neutrinos!
In his Nobel Prize lecture (2002), Raymond Davis says
“The solar neutrino problem lasted from 1967–2001. Over this period
neither the measured flux nor the predicted flux changed significantly.
I never found anything wrong with my experiment. John Bahcall never
found anything wrong with the standard solar model.”
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There were several possible explanations for this.
1. Nuclear fusion is not taking place in the Sun (but other evidence suggests it
is)
2. The experiment was not calibrated properly (but several other experiments
have detected the same result)
3. The rate of fusion inside the Sun is lower than we thought (but we know
how much energy has to come out!)
4. Something is happening to some of the neutrinos on the way so we
can’t detect them.
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In the late 1990s a new detector
called Super-Kamiokande was built in
Japan, using 50,000 tonnes of pure
water and 11,200 photomultiplier
tubes in a 40m high x 40m diameter
cylinder.
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Oscillating neutrinos
In 1998 a Super-Kamiokande team announced they had detected
evidence of neutrinos oscillating between different varieties, only one
of which – the electron neutrino – can be detected.
This not only explains the results of the solar neutrino experiment, but
also implies that neutrinos have mass. This in turn has enormous
implications for the total mass and hence the fate of the universe.
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“What appliance can pierce through the outer layers of
a star and test the conditions within?"
– Arthur Eddington
HST images of Betelgeuse and Mira
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What is a star?
“... a gravitationally confined thermonuclear reactor whose composition
evolves as energy is lost to radiation and neutrinos”
– Woosley, Heger and Weaver 2002
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Next week
...we’ll look at our own Sun, and what we have learned when we observe
a star really up close and personal
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Further reading
• Most books which discuss stars have a good discussion of how stars work and what makes them shine. The books by James
Kaler I suggested in Lecture 1 have a good description; so do several of the books I’ll be recommending in upcoming
lectures, such as “Cosmic Catastrophes: Exploding stars, Black holes, and Mapping the Universe” by J. Craig Wheeler
(Cambridge UP, 2007).
• John Bahcall, one of the giants of solar neutrino research, has an article on the neutrino problem at the Nobel Prize
website called “Solving the Mystery of the Missing Neutrinos”, http://nobelprize.org/physics/articles/bahcall/index.html
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Sources for images used:
• SOHO image of a solar prominence on the Sun: from SOHO: Best of SOHO Images http://sohowww.nascom.nasa.gov/
• HST images of Betelgeuse and Mira: from the Hubble News Center: http://hubblesite.org/newscenter/newsdesk/archive/releases/1996/04/ and
http://hubblesite.org/newscenter/newsdesk/archive/releases/1997/26/
• Four forces: from “All Four Engines Out: Volcanic Ash, St Elmo’s Fire and Electrostatics” by David Jamieson http://www.ph.unimelb.edu.au/~dnj/jl/jl99/index.htm
• Structure of atom: from “Introductory Chemistry” by Nivaldo J. Tro, Fig. 04-06, http://wps.prenhall.com/wps/media/objects/476/488316/ch04.html
• Strength of the electromagnetic force: from “All Four Engines Out: Volcanic Ash, St Elmo’s Fire and Electrostatics” by David Jamieson,
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http://www.ph.unimelb.edu.au/~dnj/jl/jl99/index.htm
Mass-luminosity relation: from Auckland Astronomical Society http://www.astronomy.org.nz/aas/MonthlyMeetings/MeetingMay2002.asp
Proton-proton chain: from Astronomy 162: Stars, Galaxies and Cosmology: The Proton-Proton Chain http://csep10.phys.utk.edu/astr162/lect/energy/ppchain.html
CNO cycle: from “Formation of the High Mass Elements (a/k/a What Happens Inside a Star)” http://aether.lbl.gov/www/tour/elements/stellar/CNO.html
Equilibrium: from Chandra resources http://chandra.harvard.edu/resources/illustrations/normalstars.html
Brown dwarf Gliese 229B: from Hubble News Center, http://hubblesite.org/newscenter/newsdesk/archive/releases/1995/48/
Eta Carinae: from Astronomy Picture of the Day 2002 April 28, http://antwrp.gsfc.nasa.gov/apod/ap020428.html
Sun’s temperature structure: from Astronomy 122 Mid-term Exam 2 Review, http://physics.uoregon.edu/~jimbrau/astr122/Notes/Exam2rev.html
Convection: from Chandra Resource: Normal stars (Illustrations), http://chandra.harvard.edu/resources/illustrations/normalstars.html
Comparison of energy transport in different types of stars: from “Introduction to Astronomy” by Bram Borosn, http://www.bramboroson.com/astro/apr3.html
Solar granules: from “Solar Physics: Photospheric features” http://science.nasa.gov/ssl/pad/solar/feature1.htm
Davis experiment: from “Big World of Small Neutrinos”, http://conferences.fnal.gov/lp2003/forthepublic/neutrinos/
Super-Kamiokande: Diagram: from Super-K at U. of Washington, http://www.phys.washington,edu/~superk/uwgroup.html
Photo of interior: from Photo Album of the SK detector, http://www-sk.icrr.u-tokyo.ac.jp/doc/sk/photo/index.html
Solar Dynamics Observatory image of the Sun: from http://sdo.gsfc.nasa.gov/gallery/main/item/151
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